Non-Equilibrium Transport Through a Gate-Controlled Barrier on the Quantum Spin
Hall Edge
Roni Ilan,1, ∗ J´rˆme Cayssol,1, 2, 3 Jens H. Bardarson,1, 4 and Joel E. Moore1, 4
eo
1
arXiv:1206.5211v1 [cond-mat.str-el] 22 Jun 2012
2
Department of Physics, University of California, Berkeley, California 94720, USA
Max-Planck-Institut f¨r Physik Komplexer Systeme, N¨thnitzer Str. 38, 01187 Dresden, Germany
u
o
3
LOMA (UMR-5798), CNRS and University Bordeaux 1, F-33045 Talence, France
4
Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720
The Quantum Spin Hall insulator is characterized by the presence of gapless helical edge states
where the spin of the charge carriers is locked to their direction of motion. In order to probe the
properties of the edge modes, we propose a design of a tunable quantum impurity realized by a
local gate under an external magnetic ﬁeld. Using the integrability of the impurity model, the
conductance is computed for arbitrary interactions, temperatures and voltages, including the eﬀect
of Fermi liquid leads. The result can be used to infer the strength of interactions from transport
experiments.
The quantum spin Hall eﬀect (QSHE) is a property
of certain two-dimensional electron systems with strong
spin-orbit coupling [1, 2]. The bulk of the system is electrically insulating, while a conducting “helical edge” exists at the boundary in which electrons of opposite spin
move in opposite directions [3–5]. Due to this reduction
of the number of degrees of freedom, the QSHE edge is
expected to realize the physics of a spinless Luttinger liquid, as opposed to a conventional one-dimensional wire
that represents a spinful Luttinger liquid [6]. The Luttinger liquid is the generic state of metallic interacting
electrons in one dimension [7], while metallic electrons in
higher dimensions typically form a Fermi liquid.
The QSHE is realized in (Hg,Cd)Te quantum wells [8,
9] where measurements of the conductance indicate the
existence of helical edge modes. The simplest measurement to perform on such a system would be a twoterminal conductance measurement. Such a measurement can conﬁrm that the current is carried by helical
one-dimensional edge channels, but it can neither provide information on the interaction strength within those
channels, nor verify the expected Luttinger liquid behavior. This is the case because, when a clean interacting
wire is placed between Fermi liquid contacts (modeled
as a non-interacting wires), the measured conductance is
insensitive to the interactions [10, 11].
As it turns out, there is a way in which the two terminal conductance can provide information on the interaction strength within the edge modes. A common
way of studying one-dimensional systems, both theoretically and experimentally, is by exploring impurity eﬀects
on measurable quantities such as their conductance. In
general, the problem becomes quite involved when interactions are present, and one usually has to rely on
the asymptotic behavior of such quantities (at high or
low temperatures, for example) to extract information
on the interaction strength. However, in some unique
cases certain properties of the edge model make it possible to obtain exact solutions. The QSHE edge is an
example of such a system, since the model of a spinless
Luttinger liquid with an impurity is “integrable” [12].
In order to utilize the powerful tool of integrability to
describe actual measurements on a QSHE edge, backscattering must be induced within a single edge (the model
describing backscattering between the two edges of the
QSHE system is not integrable). In principle, this can be
done by means of a magnetic impurity that locally breaks
time-reversal symmetry. However, it is much more desirable to ﬁnd a way to engineer an impurity with a tunable
strength, in order to induce the crossover between weak
and strong backscattering.
In this work we consider combining the eﬀects of an
externally applied magnetic ﬁeld and a local gate voltage
to form an artiﬁcial impurity on the QSHE edge. The
magnetic ﬁeld direction is carefully chosen such that it
breaks time-reversal symmetry yet leaves the edge modes
gapless. These edge modes, now unprotected, become
sensitive to the local perturbation generated by the gate
in the form of an induced Rashba spin-orbit coupling.
The strength of the impurity is set both by the magnetic ﬁeld and the gate voltage. With controlled means
for introducing an impurity, the integrality of the edge
model [12–14] allows us to extract the shape of the nonequilibrium, ﬁnite temperature conductance curve, which
strongly depends on the value of the Luttinger parameter. Hence measuring the conductance throughout the
crossover from weak to strong backscattering could provide information on the interaction strength within the
edge channels.
The setup we have in mind (see Fig. 1) is similar in
spirit to a quantum point contact in fractional quantum
Hall eﬀect (FQHE) devices [15]. There, particle backscattering between modes with opposite chirality is enhanced
with the aid of two gates depleting the electron density
and bringing the two edges of the sample closer together.
However, for the QSHE device we consider, backscattering between counter-propagating modes takes place on
the same edge. Hence, we do not require that the two
2
Hamiltonian
H = −i vF σz ∂x + µB
FIG. 1: Schematic diagram of the proposed experimental
setup. Voltage the top gate is used to locally tune the strength
of the Rashba spin-orbit coupling. Combined with a magnetic
ﬁeld aligned along the electron spin quantization axis, a gap
appears in the edge spectrum, giving rise to backscattering in
the helical edge.
edges of the sample be brought together, and a single
gate is suﬃcient. Recently, leading corrections to the linear conductance induced by a generic magnetic impurity
in a fractional topological insulator were calculated [16].
There, unlike the integer case we study, an edge with
repulsive interaction can be stable to magnetic perturbations.
Note that although both the QSHE and the FQHE
edges realize a spinless Luttinger liquid, the Luttinger parameter for the QSHE can in principle obtain any value,
while for the FQHE it is restricted to quantized values.
Another crucial diﬀerence between the two systems is
embodied in the eﬀect of Fermi liquid contacts discussed
earlier. For the FQHE, contacts are expected to have no
eﬀect on the conductance, due to the spatial separation
of modes of opposite chirality. This has been observed
in experiments [17–19]. Therefore, the QSHE case has
the potential to provide the ﬁrst experimental test of integrability at non-quantized values of the Luttinger parameter and in the process verify the eﬀects associated
with Fermi-liquid contacts.
We start by considering the non-interacting case, solving the scattering problem of two gapless regions separated by a ﬁnite strip in which an energy gap is present.
We ﬁnd the reﬂection strength and show that it can display resonant behavior for some values of the parameters.
We then consider interactions and use a method known
as the thermodynamic Bethe ansatz to obtain the nonequilibrium ﬁnite temperature conductance for various
values of the Luttinger parameter [13].
The low energy physics of the non-interacting edge in
the presence of a magnetic ﬁeld B and a position dependent Rashba spin-orbit coupling α(x) is described by the
ge
i
B · σ − {α, ∂x }σy ,
2
2
(1)
where vF is the Fermi velocity, the σ’s are the Pauli matrices, {., .} denotes an anticommutator, ge is the electron
Land´ g-factor and µB is the Bohr magneton. To sime
plify notation, in the following we take = 1 and deﬁne
M = µB ge B/2. For M = α = 0 the spectrum of this
Hamiltonian is gapless, E = ±vF p. When a magnetic
ﬁeld is turned on, the energy spectrum becomes gapped,
unless the magnetic ﬁeld is parallel to the spin quantization axis of the electron. In that case the eﬀect of the ﬁeld
is merely to shift the Dirac point and E = ±(vF p + M ).
In the absence of a magnetic ﬁeld, a ﬁnite constant spinorbit interaction α(x) = α0 renormalizes the electron ve2
2
locity to vα = α0 + vF , and rotates the electron spin
quantization axis by an angle cos θ = vF /vα about the x
axis [20]. Note that the spins of the counter-propagating
modes remain anti-parallel in the presence of the Rashba
term as required by time-reversal symmetry.
Let us now consider a system in which the magnetic
ﬁeld is uniform and points along the spin quantization
axis, while a ﬁnite constant Rashba coupling exists only
within a ﬁnite strip of width d, α(x) = α0 Θ(x)Θ(d − x).
Outside the strip (x < 0, x > d), the energy spectrum is
gapless, while within the strip (0 < x < d), the external
magnetic ﬁeld is no longer aligned with the spin polarization axis, and the energy spectrum becomes gapped
2
2
2
E = ± (vα p + vF M/vα )2 + α0 M 2 /vα ,
(2)
with the energy gap Eg = |2α0 M/vα |. In the presence
of the external ﬁeld, the two otherwise decoupled spinors
now mix in the region combining both the ﬁeld and the
spin-orbit coupling. The result is a square scattering barrier, from which incoming waves can be reﬂected. In
the limit of a narrow constriction, this region acts as a
localized impurity in our helical quantum wire, whose
strength is controlled by M and α0 . In reality this can
be realized by varying the voltage of a nearby electrostatic gate which enhances the Rashba coupling in the
vicinity of the gate, while the Rashba coupling far from
the gate is negligible [21].
We solve the scattering problem by deﬁning the scattering state in each region to be
ψR eipR x + rψL eipL x
x<0
ip+ x
ip− x
Ψ(x) = a+ ψ+ e
+ a− ψ− e
0d
where ψR = (1, 0), ψL = (0, 1) and ψ± = (iαp± , vF p± −
M − E). The momenta pR/L = (±E − M )/vF correspond to the right (R) and left (L) movers outside the
strip, while p± are the two momenta inside the strip,
corresponding to the solutions of (2) at a given energy.
3
Tx1 ,x0 = Px e
i
x1
x0
dx
(vF σz +ασy )
v 2 +α2
F
i
[E+M σz + 2 (∂x α)σy ]
1.0
E/E0
3
10
15
0.8
0.6
R
The nontrivial part of the solution for r and t, the reﬂection and transmission amplitudes is to ﬁnd the correct
matching condition for the wave function Ψ at x = 0, d.
For a general proﬁle of α(x), the Schroedinger equation
Hψ = Eψ [Eq. (1)] can be solved formally as Ψ(x1 ) =
Tx1 ,x0 Ψ(x0 ) where the transfer matrix is written as
0.4
,
(3)
0.2
with Px representing the path ordering operator. For a
step in α, α(x) = α0 Θ(x), we set x0 = −δ and x1 = δ
and then take the limit of δ → 0. The contribution of the
terms including the magnetic ﬁeld and the energy in the
exponent will vanish, and we are left with the matching
condition
ψ(0+ ) =
vF
vα
0.0
0
50
100
M/E0
150
200
FIG. 2: Reﬂection probability R as a function of the normalized magnetic ﬁeld M for various Fermi energies E, and
α0 /vF = 0.1. The energy unit is E0 = vF /d.
1/2
eiθ0 σx ψ(0− ),
(4)
with tan 2θ0 = α0 /vF .
Using the boundary condition (4) at x = 0, and a similar one at x = d, we obtain the solutions for r, t. The
analytical form of these solutions is lengthy, therefore
we will not present it here but rather plot the reﬂection
probability R = |r|2 in Fig. 2. The behavior of R as a
function of the ﬁeld at a ﬁxed (nonzero) value of α0 /vF
can be described as follows: at zero ﬁeld, the reﬂection
is zero, while for large ﬁelds M
vα E/α0 , the reﬂection
is perfect, R = 1. In between, R can display two types
of behaviors, depending on whether the waves inside the
barrier are evanescent or propagating. For evanescent
2
2
waves E 2 < α0 M 2 /vα , R rises monotonically towards
unity, while propagating waves result in an oscillating
reﬂection amplitude, due to Fabry-P´rot type of interfere
ence resonances. The condition for a resonance is simply
(p+ − p− )d = 2πn, and depends both on the value of
α0 and M . Such Fabry-P´rot resonances have also been
e
predicted for a QSHE edge state under a spatially inhomogeneous magnetic ﬁeld [22].
While the resonances appearing for the ﬁnite barrier
would provide a test for the existence of helical edge
modes in the absence of interactions, it is expected that
interaction eﬀects are important in 1D systems and can
renormalize drastically the backscattering created by a
single impurity [23]. This is true even for weak repulsive interactions since single electron backscattering is
described by a relevant operator (in the renormalization group sense), leading to a crossover from a weakbackscattering regime to a strong backscattering regime
as the temperature is lowered. The non-interacting solution is still useful in estimating the bare backscattering
strength and its dependence on α0 and M . In our discussion of the non-interacting problem we considered a
region of ﬁnite width d as the scatterer. In order to calculate conductance in the interacting case we need to
consider a point like scatterer. The bare backscattering
strength for such an impurity could be estimated from
our previous calculation by taking the limit of a very
narrow barrier [30]. For weak magnetic ﬁelds and small
2
α0 it is simply given by R ∼ (M α2k /vF )2 , where α2k is
the 2k component of the Fourier decomposition of α(x).
The Hamiltonian of the interacting QSHE edge in the
bosonization language is
H=
v
4πg
dx(∂x φR )2 + (∂x φL )2 ,
(5)
where φR/L are left moving and right moving boson ﬁelds,
g is the Luttinger liquid parameter and v is the edge velocity renormalized by interactions [24]. A backscattering
term couples to φL − φR
HB = λ cos(φL (0) − φR (0)).
(6)
By deﬁning even and odd non-local combinations of the
√
ﬁelds φe/o = 1/ 2(φL (x, t)±φR (−x, t)), the backscattering term couples only to φo , and the Hamiltonian breaks
into two decoupled contributions. The part of the Hamiltonian describing the odd ﬁelds is integrable, since it
is identical to the massless limit of the boundary sineGordon (SG) model [12, 14]. The even ﬁeld theory is
free and does not interact with the impurity.
The integrability of the SG model was previously used
by Fendley, Ludwig, and Saleur (FLS) [12, 14] to calculate the non-linear conductance in a point contact geometry for fractional quantum Hall states at ν = 1/m. In
Ref. 25 a similar formula is derived for the conductance
at ν = 1 − 1/m by exploiting the relation between the
Kondo problem in a magnetic ﬁeld and the SG model at
ﬁnite voltage. Here, we use the same method to compute
the diﬀerential conductance curves at m = 3, 4, 5 in order
to demonstrate the behavior of the QSHE edge transport
at 1/2 < g < 1. A result for continuously variable g can
in principle be computed using a more involved technique
developed in [26]. Also, if the repulsive interactions are
4
ln
dθ
× (7)
2
cosh [θ − ln(TB /T )]
1 + e(m−1)V /2T − + (θ) 1 + e−(m−1)V /2T − + (∞)
1 + e−(m−1)V /2T − + (θ) 1 + e(m−1)V /2T − + (∞)
.
Here TB is an energy scale related to the impurity
strength λ by TB = Cλ1/(1−g) [14, 25], where C is a nonuniversal (cutoﬀ dependent) constant, + is the quasienergy of the kink solution of the sine-Gordon model,
and θ the rapidity. The energy + (θ) of the kinks is
computed numerically by solving a set of coupled integral equations obtained from the thermodynamic Bethe
ansatz. The full details of these equations are given in
supplementary material.
To account for the eﬀect of non-interacting leads on
the calculation of the conductance, we adapt a result
from Ref. 28, where a self-consistency condition was derived for the chemical potential of the various excitations
inside the wire, which is not equal to the external applied
voltage. The consequences of this self-consistency condition were extensively explored in Ref. 28 for g = 1/m.
Here we carry out a similar analysis for g = 1 − 1/m.
Denoting the chemical potential for the kinks and antikinks by µ± = ±(m − 1)W/2 and the external voltage by
V , the self-consistency condition for W is [28]
V =−
e2
h
1−
1
g
0.6
0.4
0.2
V /T = 1.0
0.0 −4
10
1.0
0.8
10−3
10−2
10−1
100
TB /T
101
102
103
m
3
4
5
0.6
0.4
0.2
TB /T = 0.5
0.0 −1
10
1.0
0.8
100
101
102
103
104
V /T
m
3
4
5
0.6
0.4
0.2
I(W ) + W.
(8)
The results for the diﬀerential conductance G = dI/dV ,
with and without the contact correction, are presented
in Fig. 3. The asymptotic behavior of G as a function
of TB /T matches the known predictions [23], namely
2/g−2
G
e2 /h (T /TB )
at low temperature, and G
2(1−g)
G/(e2/h)
T (m − 1)
2
m
3
4
5
0.8
G/G∞
I(V, TB , T ) =
1.0
G/G∞
weak (g close to 1), a more direct approach based on resummed perturbation theory [27] can be used, but it is
currently unclear whether interactions are weak in the
QSHE edge.
For g = 1 − 1/m, the current along the QSHE edge is
given by [25]:
e2 /h 1 − (TB /T )
at high temperature.
Though we have computed the full curve for particular
values of g, all curves show similar features and a comparison with experimental data should conﬁrm the expected
Luttinger liquid behavior and yield a good estimate for
the value of g. Note that when contact corrections are included the conductance always saturates to e2 /h at high
voltage or in the absence of backscattering. Nevertheless,
the curve shape itself highly depends on g, and in particular, the exponents of the asymptotic behavior remain
the same as without the correction.
The feasibility of our proposal depends both on the
stability of the QSHE edge in presence of a magnetic
ﬁeld and the spin direction of the modes. The behavior
of the QSHE under a magnetic ﬁeld has been studied in
TB /T = 2.0
0.0 −1
10
100
101
102
103
104
V /T
FIG. 3: Diﬀerential conductance for diﬀerent values of g =
1 − 1/m. Top ﬁgure: G = dI/dV (with contact corrections)
as a function of TB /T . Bottom ﬁgures: G = dI/dV as a
function of V /T for two diﬀerent values of TB /T . The diﬀerential conductance has been scaled with the value G∞ = e2 /h
and (1 − 1/m)e2 /h with and without contact corrections, respectively. The curves without contact corrections have been
shifted down by 0.3 for clarity.
an experiment where the conductance was measured for
various tilt angles of the ﬁeld with respect to the plane of
the 2D electron gas [2, 8, 29]. The results show that on
top of the contribution from the Zeeman coupling, when
the ﬁeld is perpendicular to the plane, the conductance
drops rapidly with the ﬁeld strength due to orbital eﬀects.
Nevertheless, a peak in the conductance of typical width
B = 10mT exists at T = 30mK. Orbital eﬀects result in
an eﬀective g-factor values of 20 − 50, the typical Fermi
velocity is estimated to be vF = 5.5 × 105 m/s, and α0 ≈
5
5 × 104 m/s. Therefore, even under the most restrictive
conditions one can obtain a gap size of Eg ≈ 100−300mK
in the vicinity of the gate in our setup.
In our analysis we have made the assumption that the
spin quantization axis far from the gated region is ﬁxed
along the edge, and therefore it is possible to align the
magnetic ﬁeld such that it does not gap out the edge
modes in those regions. In principle, the preferred spin
quantization axis along the edge is determined by the
properties of the material, is not protected, and may tilt
in a complicated way along the edge due to ﬂuctuations
of the Rashba coupling. However, if the edge is made
smooth enough it is reasonable to assume that ﬂuctuations in the Rashba coupling have a much smaller eﬀect
on the edge states than the intentional coupling induced
by the gate, and therefore our analysis remains valid.
To summarize, in this paper we proposed a realistic
and controlled setup for transport experiments on the
QSHE edge using the combined eﬀect of time-reversal
symmetry breaking and induced spin-orbit coupling. Using the integrability of the resulting edge model we predict the form of the non-linear conductance, which could
be compared with future experimental data.
We would like to thank Paul Fendley for useful
discussions. The authors acknowledge support from
AFOSR MURI (RI), the ONR EU/FP7 under contract TEMSSOC and from ANR through project 2010BLANC-041902 (ISOTOP) (JC), the Nanostructured
Thermoelectrics program of LBNL (JHB), and DARPA
(JEM).
∗
Electronic address: rilan@berkeley.edu
[1] M. Hasan and C. Kane, Rev. Mod. Phys. 82, 3045 (2010).
[2] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057
(2011).
[3] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801
(2005).
[4] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802
(2005).
[5] B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Science
314, 1757 (2006).
[6] T. Giamarchi, Quantum Physics in One Dimension (Oxford University Press, 2004).
[7] F. D. M. Haldane, J. Phys. C: Solid State Phys. 14, 2585
(2000).
[8] M. Konig, S. Wiedmann, C. Brune, A. Roth, H. Buhmann, L. W. Molenkamp, X. L. Qi, and S. C. Zhang,
Science 318, 766 (2007).
[9] A. Roth, C. Brune, H. Buhmann, L. W. Molenkamp,
J. Maciejko, X. L. Qi, and S. C. Zhang, Science 325, 294
(2009).
[10] D. Maslov and M. Stone, Phys. Rev. B 52, R5539 (1995).
[11] I. Saﬁ and H. Schulz, Phys. Rev. B 52, R17040 (1995).
[12] P. Fendley, A. Ludwig, and H. Saleur, Phys. Rev. Lett.
74, 3005 (1995).
[13] H. Saleur, in Topological aspects of low dimensional sys-
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
tems - Les Houches 1998 summer school session LXIX,
edited by A. Comtet, T. Jolicoeur, and S. Ouvry (Nato
advanced study institute, 1998).
P. Fendley, A. Ludwig, and H. Saleur, Phys. Rev. B 52,
8934 (1995).
A. Chang, Rev. Mod. Phys. 75, 1449 (2003).
B. B´ri and N. Cooper, Phys. Rev. Lett. 108, 206804
e
(2012).
F. P. Milliken, C. P. Umbach, and R. A. Webb, Solid
State Commun. 97, 309 (1996).
X. Du, I. Skachko, F. Duerr, A. Luican, and E. Y. Andrei,
Nature 462, 192 (2009).
K. I. Bolotin, F. Ghahari, M. D. Shulman, H. L. Stormer,
and P. Kim, Nature 462, 196 (2009).
J. V¨yrynen and T. Ojanen, Phys. Rev. Lett. 106,
a
076803 (2011).
J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys.
Rev. Lett. 78, 1335 (1997).
A. Soori, S. Das, and S. Rao, arXiv:1112.5400v1.
C. Kane and M. Fisher, Phys. Rev. Lett. 68, 1220 (1992).
J. Teo and C. Kane, Phys. Rev. B 79, 235321 (2009).
P. Fendley, F. Lesage, and H. Saleur, J. Stat. Phys. 85,
211 (1996).
P. Dorey and R. Tateo, Nucl. Phys. B 563, 573 (1999).
D. Yue, L. Glazman, and K. Matveev, Phys. Rev. B 49,
1966 (1994).
A. Koutouza, F. Siano, and H. Saleur, J. Phys. A: Math.
Gen. 34, 5497 (2001).
M. K¨nig, H. Buhmann, L. W. Molenkamp, T. Hughes,
o
C.-X. Liu, X.-L. Qi, and S.-C. Zhang, J. Phys. Soc. Jpn.
77, 031007 (2008).
Strictly speaking, in the limit of zero width the transmission for this problem becomes unity.
6
Supplementary material: Thermodynamic Bethe
Ansatz equations
The thermodynamic Bethe ansatz equations relevant
for our particular model are the following coupled integral equations [25]:
j (θ)/T
= δj1 eθ −
Njk
k
−(
Lk (θ ) = ln 1 + e
k (θ
excitations of positive and negative charge respectively,
while the breathers are neutral bound states of a kink
and an anti-kink, and therefore their chemical potential
is alway set to zero. In general, + = − , and for ν =
1 − 1/m, the matrix element Njk = 1 if the nodes j and
k are connected in the following diagram
1
dθ
Lk (θ ),
2π cosh(θ − θ )
)+µk )/T
.
(A.9)
In this set of coupled integral equations, the indices j and
k label the diﬀerent excitation of the SG model. There
are three types of excitations: breathers (1, .., m − 2),
kinks (+) and the antikinks (−). The functions k label
their energies (parametrized by an angle θ known as the
rapidity). Kinks and anti-kinks are the charge carrying
otherwise Njk = 0.
We solve these equation numerically for arbitrary values of m. Then the energy of the kinks is used in Eq. (7)
to compute the conductance without contact correction,
or combined with the self-consistency condition deﬁned
in Eq.(8), to evaluate the conductance in presence of contact corrections.